// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2009 Gael Guennebaud <gael.guennebaud@inria.fr>
//
// This Source Code Form is subject to the terms of the Mozilla
// Public License v. 2.0. If a copy of the MPL was not distributed
// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.

#ifndef EIGEN_AUTODIFF_SCALAR_H
#define EIGEN_AUTODIFF_SCALAR_H

namespace Eigen {

namespace internal {

    template <typename A, typename B> struct make_coherent_impl
    {
        static void run(A&, B&) {}
    };

    // resize a to match b is a.size()==0, and conversely.
    template <typename A, typename B> void make_coherent(const A& a, const B& b)
    {
        make_coherent_impl<A, B>::run(a.const_cast_derived(), b.const_cast_derived());
    }

    template <typename DerivativeType, bool Enable> struct auto_diff_special_op;

}  // end namespace internal

template <typename DerivativeType> class AutoDiffScalar;

template <typename NewDerType> inline AutoDiffScalar<NewDerType> MakeAutoDiffScalar(const typename NewDerType::Scalar& value, const NewDerType& der)
{
    return AutoDiffScalar<NewDerType>(value, der);
}

/** \class AutoDiffScalar
  * \brief A scalar type replacement with automatic differentiation capability
  *
  * \param DerivativeType the vector type used to store/represent the derivatives. The base scalar type
  *                 as well as the number of derivatives to compute are determined from this type.
  *                 Typical choices include, e.g., \c Vector4f for 4 derivatives, or \c VectorXf
  *                 if the number of derivatives is not known at compile time, and/or, the number
  *                 of derivatives is large.
  *                 Note that DerivativeType can also be a reference (e.g., \c VectorXf&) to wrap a
  *                 existing vector into an AutoDiffScalar.
  *                 Finally, DerivativeType can also be any Eigen compatible expression.
  *
  * This class represents a scalar value while tracking its respective derivatives using Eigen's expression
  * template mechanism.
  *
  * It supports the following list of global math function:
  *  - std::abs, std::sqrt, std::pow, std::exp, std::log, std::sin, std::cos,
  *  - internal::abs, internal::sqrt, numext::pow, internal::exp, internal::log, internal::sin, internal::cos,
  *  - internal::conj, internal::real, internal::imag, numext::abs2.
  *
  * AutoDiffScalar can be used as the scalar type of an Eigen::Matrix object. However,
  * in that case, the expression template mechanism only occurs at the top Matrix level,
  * while derivatives are computed right away.
  *
  */

template <typename DerivativeType>
class AutoDiffScalar
    : public internal::auto_diff_special_op<
          DerivativeType,
          !internal::is_same<typename internal::traits<typename internal::remove_all<DerivativeType>::type>::Scalar,
                             typename NumTraits<typename internal::traits<typename internal::remove_all<DerivativeType>::type>::Scalar>::Real>::value>
{
public:
    typedef internal::auto_diff_special_op<
        DerivativeType,
        !internal::is_same<typename internal::traits<typename internal::remove_all<DerivativeType>::type>::Scalar,
                           typename NumTraits<typename internal::traits<typename internal::remove_all<DerivativeType>::type>::Scalar>::Real>::value>
        Base;
    typedef typename internal::remove_all<DerivativeType>::type DerType;
    typedef typename internal::traits<DerType>::Scalar Scalar;
    typedef typename NumTraits<Scalar>::Real Real;

    using Base::operator+;
    using Base::operator*;

    /** Default constructor without any initialization. */
    AutoDiffScalar() {}

    /** Constructs an active scalar from its \a value,
        and initializes the \a nbDer derivatives such that it corresponds to the \a derNumber -th variable */
    AutoDiffScalar(const Scalar& value, int nbDer, int derNumber) : m_value(value), m_derivatives(DerType::Zero(nbDer))
    {
        m_derivatives.coeffRef(derNumber) = Scalar(1);
    }

    /** Conversion from a scalar constant to an active scalar.
      * The derivatives are set to zero. */
    /*explicit*/ AutoDiffScalar(const Real& value) : m_value(value)
    {
        if (m_derivatives.size() > 0)
            m_derivatives.setZero();
    }

    /** Constructs an active scalar from its \a value and derivatives \a der */
    AutoDiffScalar(const Scalar& value, const DerType& der) : m_value(value), m_derivatives(der) {}

    template <typename OtherDerType>
    AutoDiffScalar(
        const AutoDiffScalar<OtherDerType>& other
#ifndef EIGEN_PARSED_BY_DOXYGEN
        ,
        typename internal::enable_if<internal::is_same<Scalar, typename internal::traits<typename internal::remove_all<OtherDerType>::type>::Scalar>::value &&
                                         internal::is_convertible<OtherDerType, DerType>::value,
                                     void*>::type = 0
#endif
        )
        : m_value(other.value()), m_derivatives(other.derivatives())
    {
    }

    friend std::ostream& operator<<(std::ostream& s, const AutoDiffScalar& a) { return s << a.value(); }

    AutoDiffScalar(const AutoDiffScalar& other) : m_value(other.value()), m_derivatives(other.derivatives()) {}

    template <typename OtherDerType> inline AutoDiffScalar& operator=(const AutoDiffScalar<OtherDerType>& other)
    {
        m_value = other.value();
        m_derivatives = other.derivatives();
        return *this;
    }

    inline AutoDiffScalar& operator=(const AutoDiffScalar& other)
    {
        m_value = other.value();
        m_derivatives = other.derivatives();
        return *this;
    }

    inline AutoDiffScalar& operator=(const Scalar& other)
    {
        m_value = other;
        if (m_derivatives.size() > 0)
            m_derivatives.setZero();
        return *this;
    }

    //     inline operator const Scalar& () const { return m_value; }
    //     inline operator Scalar& () { return m_value; }

    inline const Scalar& value() const { return m_value; }
    inline Scalar& value() { return m_value; }

    inline const DerType& derivatives() const { return m_derivatives; }
    inline DerType& derivatives() { return m_derivatives; }

    inline bool operator<(const Scalar& other) const { return m_value < other; }
    inline bool operator<=(const Scalar& other) const { return m_value <= other; }
    inline bool operator>(const Scalar& other) const { return m_value > other; }
    inline bool operator>=(const Scalar& other) const { return m_value >= other; }
    inline bool operator==(const Scalar& other) const { return m_value == other; }
    inline bool operator!=(const Scalar& other) const { return m_value != other; }

    friend inline bool operator<(const Scalar& a, const AutoDiffScalar& b) { return a < b.value(); }
    friend inline bool operator<=(const Scalar& a, const AutoDiffScalar& b) { return a <= b.value(); }
    friend inline bool operator>(const Scalar& a, const AutoDiffScalar& b) { return a > b.value(); }
    friend inline bool operator>=(const Scalar& a, const AutoDiffScalar& b) { return a >= b.value(); }
    friend inline bool operator==(const Scalar& a, const AutoDiffScalar& b) { return a == b.value(); }
    friend inline bool operator!=(const Scalar& a, const AutoDiffScalar& b) { return a != b.value(); }

    template <typename OtherDerType> inline bool operator<(const AutoDiffScalar<OtherDerType>& b) const { return m_value < b.value(); }
    template <typename OtherDerType> inline bool operator<=(const AutoDiffScalar<OtherDerType>& b) const { return m_value <= b.value(); }
    template <typename OtherDerType> inline bool operator>(const AutoDiffScalar<OtherDerType>& b) const { return m_value > b.value(); }
    template <typename OtherDerType> inline bool operator>=(const AutoDiffScalar<OtherDerType>& b) const { return m_value >= b.value(); }
    template <typename OtherDerType> inline bool operator==(const AutoDiffScalar<OtherDerType>& b) const { return m_value == b.value(); }
    template <typename OtherDerType> inline bool operator!=(const AutoDiffScalar<OtherDerType>& b) const { return m_value != b.value(); }

    inline const AutoDiffScalar<DerType&> operator+(const Scalar& other) const { return AutoDiffScalar<DerType&>(m_value + other, m_derivatives); }

    friend inline const AutoDiffScalar<DerType&> operator+(const Scalar& a, const AutoDiffScalar& b)
    {
        return AutoDiffScalar<DerType&>(a + b.value(), b.derivatives());
    }

    //     inline const AutoDiffScalar<DerType&> operator+(const Real& other) const
    //     {
    //       return AutoDiffScalar<DerType&>(m_value + other, m_derivatives);
    //     }

    //     friend inline const AutoDiffScalar<DerType&> operator+(const Real& a, const AutoDiffScalar& b)
    //     {
    //       return AutoDiffScalar<DerType&>(a + b.value(), b.derivatives());
    //     }

    inline AutoDiffScalar& operator+=(const Scalar& other)
    {
        value() += other;
        return *this;
    }

    template <typename OtherDerType>
    inline const AutoDiffScalar<CwiseBinaryOp<internal::scalar_sum_op<Scalar>, const DerType, const typename internal::remove_all<OtherDerType>::type>>
    operator+(const AutoDiffScalar<OtherDerType>& other) const
    {
        internal::make_coherent(m_derivatives, other.derivatives());
        return AutoDiffScalar<CwiseBinaryOp<internal::scalar_sum_op<Scalar>, const DerType, const typename internal::remove_all<OtherDerType>::type>>(
            m_value + other.value(), m_derivatives + other.derivatives());
    }

    template <typename OtherDerType> inline AutoDiffScalar& operator+=(const AutoDiffScalar<OtherDerType>& other)
    {
        (*this) = (*this) + other;
        return *this;
    }

    inline const AutoDiffScalar<DerType&> operator-(const Scalar& b) const { return AutoDiffScalar<DerType&>(m_value - b, m_derivatives); }

    friend inline const AutoDiffScalar<CwiseUnaryOp<internal::scalar_opposite_op<Scalar>, const DerType>> operator-(const Scalar& a, const AutoDiffScalar& b)
    {
        return AutoDiffScalar<CwiseUnaryOp<internal::scalar_opposite_op<Scalar>, const DerType>>(a - b.value(), -b.derivatives());
    }

    inline AutoDiffScalar& operator-=(const Scalar& other)
    {
        value() -= other;
        return *this;
    }

    template <typename OtherDerType>
    inline const AutoDiffScalar<CwiseBinaryOp<internal::scalar_difference_op<Scalar>, const DerType, const typename internal::remove_all<OtherDerType>::type>>
    operator-(const AutoDiffScalar<OtherDerType>& other) const
    {
        internal::make_coherent(m_derivatives, other.derivatives());
        return AutoDiffScalar<CwiseBinaryOp<internal::scalar_difference_op<Scalar>, const DerType, const typename internal::remove_all<OtherDerType>::type>>(
            m_value - other.value(), m_derivatives - other.derivatives());
    }

    template <typename OtherDerType> inline AutoDiffScalar& operator-=(const AutoDiffScalar<OtherDerType>& other)
    {
        *this = *this - other;
        return *this;
    }

    inline const AutoDiffScalar<CwiseUnaryOp<internal::scalar_opposite_op<Scalar>, const DerType>> operator-() const
    {
        return AutoDiffScalar<CwiseUnaryOp<internal::scalar_opposite_op<Scalar>, const DerType>>(-m_value, -m_derivatives);
    }

    inline const AutoDiffScalar<EIGEN_EXPR_BINARYOP_SCALAR_RETURN_TYPE(DerType, Scalar, product)> operator*(const Scalar& other) const
    {
        return MakeAutoDiffScalar(m_value * other, m_derivatives * other);
    }

    friend inline const AutoDiffScalar<EIGEN_EXPR_BINARYOP_SCALAR_RETURN_TYPE(DerType, Scalar, product)> operator*(const Scalar& other, const AutoDiffScalar& a)
    {
        return MakeAutoDiffScalar(a.value() * other, a.derivatives() * other);
    }

    //     inline const AutoDiffScalar<typename CwiseUnaryOp<internal::scalar_multiple_op<Real>, DerType>::Type >
    //     operator*(const Real& other) const
    //     {
    //       return AutoDiffScalar<typename CwiseUnaryOp<internal::scalar_multiple_op<Real>, DerType>::Type >(
    //         m_value * other,
    //         (m_derivatives * other));
    //     }
    //
    //     friend inline const AutoDiffScalar<typename CwiseUnaryOp<internal::scalar_multiple_op<Real>, DerType>::Type >
    //     operator*(const Real& other, const AutoDiffScalar& a)
    //     {
    //       return AutoDiffScalar<typename CwiseUnaryOp<internal::scalar_multiple_op<Real>, DerType>::Type >(
    //         a.value() * other,
    //         a.derivatives() * other);
    //     }

    inline const AutoDiffScalar<EIGEN_EXPR_BINARYOP_SCALAR_RETURN_TYPE(DerType, Scalar, product)> operator/(const Scalar& other) const
    {
        return MakeAutoDiffScalar(m_value / other, (m_derivatives * (Scalar(1) / other)));
    }

    friend inline const AutoDiffScalar<EIGEN_EXPR_BINARYOP_SCALAR_RETURN_TYPE(DerType, Scalar, product)> operator/(const Scalar& other, const AutoDiffScalar& a)
    {
        return MakeAutoDiffScalar(other / a.value(), a.derivatives() * (Scalar(-other) / (a.value() * a.value())));
    }

    //     inline const AutoDiffScalar<typename CwiseUnaryOp<internal::scalar_multiple_op<Real>, DerType>::Type >
    //     operator/(const Real& other) const
    //     {
    //       return AutoDiffScalar<typename CwiseUnaryOp<internal::scalar_multiple_op<Real>, DerType>::Type >(
    //         m_value / other,
    //         (m_derivatives * (Real(1)/other)));
    //     }
    //
    //     friend inline const AutoDiffScalar<typename CwiseUnaryOp<internal::scalar_multiple_op<Real>, DerType>::Type >
    //     operator/(const Real& other, const AutoDiffScalar& a)
    //     {
    //       return AutoDiffScalar<typename CwiseUnaryOp<internal::scalar_multiple_op<Real>, DerType>::Type >(
    //         other / a.value(),
    //         a.derivatives() * (-Real(1)/other));
    //     }

    template <typename OtherDerType>
    inline const AutoDiffScalar<EIGEN_EXPR_BINARYOP_SCALAR_RETURN_TYPE(
        CwiseBinaryOp<internal::scalar_difference_op<Scalar> EIGEN_COMMA const EIGEN_EXPR_BINARYOP_SCALAR_RETURN_TYPE(DerType, Scalar, product)
                          EIGEN_COMMA const EIGEN_EXPR_BINARYOP_SCALAR_RETURN_TYPE(typename internal::remove_all<OtherDerType>::type, Scalar, product)>,
        Scalar,
        product)>
    operator/(const AutoDiffScalar<OtherDerType>& other) const
    {
        internal::make_coherent(m_derivatives, other.derivatives());
        return MakeAutoDiffScalar(m_value / other.value(),
                                  ((m_derivatives * other.value()) - (other.derivatives() * m_value)) * (Scalar(1) / (other.value() * other.value())));
    }

    template <typename OtherDerType>
    inline const AutoDiffScalar<CwiseBinaryOp<internal::scalar_sum_op<Scalar>,
                                              const EIGEN_EXPR_BINARYOP_SCALAR_RETURN_TYPE(DerType, Scalar, product),
                                              const EIGEN_EXPR_BINARYOP_SCALAR_RETURN_TYPE(typename internal::remove_all<OtherDerType>::type, Scalar, product)>>
    operator*(const AutoDiffScalar<OtherDerType>& other) const
    {
        internal::make_coherent(m_derivatives, other.derivatives());
        return MakeAutoDiffScalar(m_value * other.value(), (m_derivatives * other.value()) + (other.derivatives() * m_value));
    }

    inline AutoDiffScalar& operator*=(const Scalar& other)
    {
        *this = *this * other;
        return *this;
    }

    template <typename OtherDerType> inline AutoDiffScalar& operator*=(const AutoDiffScalar<OtherDerType>& other)
    {
        *this = *this * other;
        return *this;
    }

    inline AutoDiffScalar& operator/=(const Scalar& other)
    {
        *this = *this / other;
        return *this;
    }

    template <typename OtherDerType> inline AutoDiffScalar& operator/=(const AutoDiffScalar<OtherDerType>& other)
    {
        *this = *this / other;
        return *this;
    }

protected:
    Scalar m_value;
    DerType m_derivatives;
};

namespace internal {

    template <typename DerivativeType> struct auto_diff_special_op<DerivativeType, true>
    //   : auto_diff_scalar_op<DerivativeType, typename NumTraits<Scalar>::Real,
    //                            is_same<Scalar,typename NumTraits<Scalar>::Real>::value>
    {
        typedef typename remove_all<DerivativeType>::type DerType;
        typedef typename traits<DerType>::Scalar Scalar;
        typedef typename NumTraits<Scalar>::Real Real;

        //   typedef auto_diff_scalar_op<DerivativeType, typename NumTraits<Scalar>::Real,
        //                            is_same<Scalar,typename NumTraits<Scalar>::Real>::value> Base;

        //   using Base::operator+;
        //   using Base::operator+=;
        //   using Base::operator-;
        //   using Base::operator-=;
        //   using Base::operator*;
        //   using Base::operator*=;

        const AutoDiffScalar<DerivativeType>& derived() const { return *static_cast<const AutoDiffScalar<DerivativeType>*>(this); }
        AutoDiffScalar<DerivativeType>& derived() { return *static_cast<AutoDiffScalar<DerivativeType>*>(this); }

        inline const AutoDiffScalar<DerType&> operator+(const Real& other) const
        {
            return AutoDiffScalar<DerType&>(derived().value() + other, derived().derivatives());
        }

        friend inline const AutoDiffScalar<DerType&> operator+(const Real& a, const AutoDiffScalar<DerivativeType>& b)
        {
            return AutoDiffScalar<DerType&>(a + b.value(), b.derivatives());
        }

        inline AutoDiffScalar<DerivativeType>& operator+=(const Real& other)
        {
            derived().value() += other;
            return derived();
        }

        inline const AutoDiffScalar<typename CwiseUnaryOp<bind2nd_op<scalar_product_op<Scalar, Real>>, DerType>::Type> operator*(const Real& other) const
        {
            return AutoDiffScalar<typename CwiseUnaryOp<bind2nd_op<scalar_product_op<Scalar, Real>>, DerType>::Type>(derived().value() * other,
                                                                                                                     derived().derivatives() * other);
        }

        friend inline const AutoDiffScalar<typename CwiseUnaryOp<bind1st_op<scalar_product_op<Real, Scalar>>, DerType>::Type>
        operator*(const Real& other, const AutoDiffScalar<DerivativeType>& a)
        {
            return AutoDiffScalar<typename CwiseUnaryOp<bind1st_op<scalar_product_op<Real, Scalar>>, DerType>::Type>(a.value() * other,
                                                                                                                     a.derivatives() * other);
        }

        inline AutoDiffScalar<DerivativeType>& operator*=(const Scalar& other)
        {
            *this = *this * other;
            return derived();
        }
    };

    template <typename DerivativeType> struct auto_diff_special_op<DerivativeType, false>
    {
        void operator*() const;
        void operator-() const;
        void operator+() const;
    };

    template <typename BinOp, typename A, typename B, typename RefType> void make_coherent_expression(CwiseBinaryOp<BinOp, A, B> xpr, const RefType& ref)
    {
        make_coherent(xpr.const_cast_derived().lhs(), ref);
        make_coherent(xpr.const_cast_derived().rhs(), ref);
    }

    template <typename UnaryOp, typename A, typename RefType> void make_coherent_expression(const CwiseUnaryOp<UnaryOp, A>& xpr, const RefType& ref)
    {
        make_coherent(xpr.nestedExpression().const_cast_derived(), ref);
    }

    // needed for compilation only
    template <typename UnaryOp, typename A, typename RefType> void make_coherent_expression(const CwiseNullaryOp<UnaryOp, A>&, const RefType&) {}

    template <typename A_Scalar, int A_Rows, int A_Cols, int A_Options, int A_MaxRows, int A_MaxCols, typename B>
    struct make_coherent_impl<Matrix<A_Scalar, A_Rows, A_Cols, A_Options, A_MaxRows, A_MaxCols>, B>
    {
        typedef Matrix<A_Scalar, A_Rows, A_Cols, A_Options, A_MaxRows, A_MaxCols> A;
        static void run(A& a, B& b)
        {
            if ((A_Rows == Dynamic || A_Cols == Dynamic) && (a.size() == 0))
            {
                a.resize(b.size());
                a.setZero();
            }
            else if (B::SizeAtCompileTime == Dynamic && a.size() != 0 && b.size() == 0)
            {
                make_coherent_expression(b, a);
            }
        }
    };

    template <typename A, typename B_Scalar, int B_Rows, int B_Cols, int B_Options, int B_MaxRows, int B_MaxCols>
    struct make_coherent_impl<A, Matrix<B_Scalar, B_Rows, B_Cols, B_Options, B_MaxRows, B_MaxCols>>
    {
        typedef Matrix<B_Scalar, B_Rows, B_Cols, B_Options, B_MaxRows, B_MaxCols> B;
        static void run(A& a, B& b)
        {
            if ((B_Rows == Dynamic || B_Cols == Dynamic) && (b.size() == 0))
            {
                b.resize(a.size());
                b.setZero();
            }
            else if (A::SizeAtCompileTime == Dynamic && b.size() != 0 && a.size() == 0)
            {
                make_coherent_expression(a, b);
            }
        }
    };

    template <typename A_Scalar,
              int A_Rows,
              int A_Cols,
              int A_Options,
              int A_MaxRows,
              int A_MaxCols,
              typename B_Scalar,
              int B_Rows,
              int B_Cols,
              int B_Options,
              int B_MaxRows,
              int B_MaxCols>
    struct make_coherent_impl<Matrix<A_Scalar, A_Rows, A_Cols, A_Options, A_MaxRows, A_MaxCols>,
                              Matrix<B_Scalar, B_Rows, B_Cols, B_Options, B_MaxRows, B_MaxCols>>
    {
        typedef Matrix<A_Scalar, A_Rows, A_Cols, A_Options, A_MaxRows, A_MaxCols> A;
        typedef Matrix<B_Scalar, B_Rows, B_Cols, B_Options, B_MaxRows, B_MaxCols> B;
        static void run(A& a, B& b)
        {
            if ((A_Rows == Dynamic || A_Cols == Dynamic) && (a.size() == 0))
            {
                a.resize(b.size());
                a.setZero();
            }
            else if ((B_Rows == Dynamic || B_Cols == Dynamic) && (b.size() == 0))
            {
                b.resize(a.size());
                b.setZero();
            }
        }
    };

}  // end namespace internal

template <typename DerType, typename BinOp> struct ScalarBinaryOpTraits<AutoDiffScalar<DerType>, typename DerType::Scalar, BinOp>
{
    typedef AutoDiffScalar<DerType> ReturnType;
};

template <typename DerType, typename BinOp> struct ScalarBinaryOpTraits<typename DerType::Scalar, AutoDiffScalar<DerType>, BinOp>
{
    typedef AutoDiffScalar<DerType> ReturnType;
};

// The following is an attempt to let Eigen's known about expression template, but that's more tricky!

// template<typename DerType, typename BinOp>
// struct ScalarBinaryOpTraits<AutoDiffScalar<DerType>,AutoDiffScalar<DerType>, BinOp>
// {
//   enum { Defined = 1 };
//   typedef AutoDiffScalar<typename DerType::PlainObject> ReturnType;
// };
//
// template<typename DerType1,typename DerType2, typename BinOp>
// struct ScalarBinaryOpTraits<AutoDiffScalar<DerType1>,AutoDiffScalar<DerType2>, BinOp>
// {
//   enum { Defined = 1 };//internal::is_same<typename DerType1::Scalar,typename DerType2::Scalar>::value };
//   typedef AutoDiffScalar<typename DerType1::PlainObject> ReturnType;
// };

#define EIGEN_AUTODIFF_DECLARE_GLOBAL_UNARY(FUNC, CODE)                                                               \
    template <typename DerType>                                                                                       \
    inline const Eigen::AutoDiffScalar<EIGEN_EXPR_BINARYOP_SCALAR_RETURN_TYPE(                                        \
        typename Eigen::internal::remove_all<DerType>::type,                                                          \
        typename Eigen::internal::traits<typename Eigen::internal::remove_all<DerType>::type>::Scalar,                \
        product)>                                                                                                     \
    FUNC(const Eigen::AutoDiffScalar<DerType>& x)                                                                     \
    {                                                                                                                 \
        using namespace Eigen;                                                                                        \
        typedef typename Eigen::internal::traits<typename Eigen::internal::remove_all<DerType>::type>::Scalar Scalar; \
        EIGEN_UNUSED_VARIABLE(sizeof(Scalar));                                                                        \
        CODE;                                                                                                         \
    }

template <typename DerType> struct CleanedUpDerType
{
    typedef AutoDiffScalar<typename Eigen::internal::remove_all<DerType>::type::PlainObject> type;
};

template <typename DerType> inline const AutoDiffScalar<DerType>& conj(const AutoDiffScalar<DerType>& x) { return x; }
template <typename DerType> inline const AutoDiffScalar<DerType>& real(const AutoDiffScalar<DerType>& x) { return x; }
template <typename DerType> inline typename DerType::Scalar imag(const AutoDiffScalar<DerType>&) { return 0.; }
template <typename DerType, typename T> inline typename CleanedUpDerType<DerType>::type(min)(const AutoDiffScalar<DerType>& x, const T& y)
{
    typedef typename CleanedUpDerType<DerType>::type ADS;
    return (x <= y ? ADS(x) : ADS(y));
}
template <typename DerType, typename T> inline typename CleanedUpDerType<DerType>::type(max)(const AutoDiffScalar<DerType>& x, const T& y)
{
    typedef typename CleanedUpDerType<DerType>::type ADS;
    return (x >= y ? ADS(x) : ADS(y));
}
template <typename DerType, typename T> inline typename CleanedUpDerType<DerType>::type(min)(const T& x, const AutoDiffScalar<DerType>& y)
{
    typedef typename CleanedUpDerType<DerType>::type ADS;
    return (x < y ? ADS(x) : ADS(y));
}
template <typename DerType, typename T> inline typename CleanedUpDerType<DerType>::type(max)(const T& x, const AutoDiffScalar<DerType>& y)
{
    typedef typename CleanedUpDerType<DerType>::type ADS;
    return (x > y ? ADS(x) : ADS(y));
}
template <typename DerType> inline typename CleanedUpDerType<DerType>::type(min)(const AutoDiffScalar<DerType>& x, const AutoDiffScalar<DerType>& y)
{
    return (x.value() < y.value() ? x : y);
}
template <typename DerType> inline typename CleanedUpDerType<DerType>::type(max)(const AutoDiffScalar<DerType>& x, const AutoDiffScalar<DerType>& y)
{
    return (x.value() >= y.value() ? x : y);
}

EIGEN_AUTODIFF_DECLARE_GLOBAL_UNARY(abs, using std::abs; return Eigen::MakeAutoDiffScalar(abs(x.value()), x.derivatives() * (x.value() < 0 ? -1 : 1));)

EIGEN_AUTODIFF_DECLARE_GLOBAL_UNARY(abs2, using numext::abs2; return Eigen::MakeAutoDiffScalar(abs2(x.value()), x.derivatives() * (Scalar(2) * x.value()));)

EIGEN_AUTODIFF_DECLARE_GLOBAL_UNARY(sqrt, using std::sqrt; Scalar sqrtx = sqrt(x.value());
                                    return Eigen::MakeAutoDiffScalar(sqrtx, x.derivatives() * (Scalar(0.5) / sqrtx));)

EIGEN_AUTODIFF_DECLARE_GLOBAL_UNARY(cos, using std::cos; using std::sin; return Eigen::MakeAutoDiffScalar(cos(x.value()), x.derivatives() * (-sin(x.value())));)

EIGEN_AUTODIFF_DECLARE_GLOBAL_UNARY(sin, using std::sin; using std::cos; return Eigen::MakeAutoDiffScalar(sin(x.value()), x.derivatives() * cos(x.value()));)

EIGEN_AUTODIFF_DECLARE_GLOBAL_UNARY(exp, using std::exp; Scalar expx = exp(x.value()); return Eigen::MakeAutoDiffScalar(expx, x.derivatives() * expx);)

EIGEN_AUTODIFF_DECLARE_GLOBAL_UNARY(log, using std::log; return Eigen::MakeAutoDiffScalar(log(x.value()), x.derivatives() * (Scalar(1) / x.value()));)

template <typename DerType>
inline const Eigen::AutoDiffScalar<EIGEN_EXPR_BINARYOP_SCALAR_RETURN_TYPE(typename internal::remove_all<DerType>::type,
                                                                          typename internal::traits<typename internal::remove_all<DerType>::type>::Scalar,
                                                                          product)>
pow(const Eigen::AutoDiffScalar<DerType>& x, const typename internal::traits<typename internal::remove_all<DerType>::type>::Scalar& y)
{
    using namespace Eigen;
    using std::pow;
    return Eigen::MakeAutoDiffScalar(pow(x.value(), y), x.derivatives() * (y * pow(x.value(), y - 1)));
}

template <typename DerTypeA, typename DerTypeB>
inline const AutoDiffScalar<Matrix<typename internal::traits<typename internal::remove_all<DerTypeA>::type>::Scalar, Dynamic, 1>>
atan2(const AutoDiffScalar<DerTypeA>& a, const AutoDiffScalar<DerTypeB>& b)
{
    using std::atan2;
    typedef typename internal::traits<typename internal::remove_all<DerTypeA>::type>::Scalar Scalar;
    typedef AutoDiffScalar<Matrix<Scalar, Dynamic, 1>> PlainADS;
    PlainADS ret;
    ret.value() = atan2(a.value(), b.value());

    Scalar squared_hypot = a.value() * a.value() + b.value() * b.value();

    // if (squared_hypot==0) the derivation is undefined and the following results in a NaN:
    ret.derivatives() = (a.derivatives() * b.value() - a.value() * b.derivatives()) / squared_hypot;

    return ret;
}

EIGEN_AUTODIFF_DECLARE_GLOBAL_UNARY(tan, using std::tan; using std::cos;
                                    return Eigen::MakeAutoDiffScalar(tan(x.value()), x.derivatives() * (Scalar(1) / numext::abs2(cos(x.value()))));)

EIGEN_AUTODIFF_DECLARE_GLOBAL_UNARY(asin, using std::sqrt; using std::asin;
                                    return Eigen::MakeAutoDiffScalar(asin(x.value()), x.derivatives() * (Scalar(1) / sqrt(1 - numext::abs2(x.value()))));)

EIGEN_AUTODIFF_DECLARE_GLOBAL_UNARY(acos, using std::sqrt; using std::acos;
                                    return Eigen::MakeAutoDiffScalar(acos(x.value()), x.derivatives() * (Scalar(-1) / sqrt(1 - numext::abs2(x.value()))));)

EIGEN_AUTODIFF_DECLARE_GLOBAL_UNARY(tanh, using std::cosh; using std::tanh;
                                    return Eigen::MakeAutoDiffScalar(tanh(x.value()), x.derivatives() * (Scalar(1) / numext::abs2(cosh(x.value()))));)

EIGEN_AUTODIFF_DECLARE_GLOBAL_UNARY(sinh, using std::sinh; using std::cosh;
                                    return Eigen::MakeAutoDiffScalar(sinh(x.value()), x.derivatives() * cosh(x.value()));)

EIGEN_AUTODIFF_DECLARE_GLOBAL_UNARY(cosh, using std::sinh; using std::cosh;
                                    return Eigen::MakeAutoDiffScalar(cosh(x.value()), x.derivatives() * sinh(x.value()));)

#undef EIGEN_AUTODIFF_DECLARE_GLOBAL_UNARY

template <typename DerType>
struct NumTraits<AutoDiffScalar<DerType>> : NumTraits<typename NumTraits<typename internal::remove_all<DerType>::type::Scalar>::Real>
{
    typedef typename internal::remove_all<DerType>::type DerTypeCleaned;
    typedef AutoDiffScalar<Matrix<typename NumTraits<typename DerTypeCleaned::Scalar>::Real,
                                  DerTypeCleaned::RowsAtCompileTime,
                                  DerTypeCleaned::ColsAtCompileTime,
                                  0,
                                  DerTypeCleaned::MaxRowsAtCompileTime,
                                  DerTypeCleaned::MaxColsAtCompileTime>>
        Real;
    typedef AutoDiffScalar<DerType> NonInteger;
    typedef AutoDiffScalar<DerType> Nested;
    typedef typename NumTraits<typename DerTypeCleaned::Scalar>::Literal Literal;
    enum
    {
        RequireInitialization = 1
    };
};

}  // namespace Eigen

namespace std {

template <typename T> class numeric_limits<Eigen::AutoDiffScalar<T>> : public numeric_limits<typename T::Scalar>
{
};

template <typename T> class numeric_limits<Eigen::AutoDiffScalar<T&>> : public numeric_limits<typename T::Scalar>
{
};

}  // namespace std

#endif  // EIGEN_AUTODIFF_SCALAR_H
